CAIRO, Feb 15 (Reuters) - Thousands of Islamists rallied in Cairo on Friday in support of Mohamed Mursi, the Muslim Brotherhood president who has been the target of protester rage in weeks of violent demonstrations.

Repeating the pattern of recent weeks, Mursi's opponents rallied again on Friday, this time gathering outside El-Quba, one of the presidential palaces in the northern suburbs of Cairo. The activists dubbed it "Checkmate Friday".

The protest, which drew several hundred people by afternoon descended into violence as night fell. State media reported that "troublemakers" had thrown rocks and petrol bombs. Security forces unleashed tear gas and water cannon, it said.

The pro-Mursi rally was called by a hardline Salafi Islamist group, Al-Gama'a al-Islamiya, in what it described as a protest under the banner "Together Against Violence".

The group waged an armed insurrection against the state in the 1990s but its leadership renounced violence more than a decade ago. It has entered mainstream politics since autocratic president Hosni Mubarak was overthrown in 2011.

The Muslim Brotherhood and its Freedom and Justice Party said it backed Friday's rally in a symbolic way but did not mobilise supporters for the event, meaning the numbers were smaller than at previous Islamist protests.

Around 60 people have been killed in Egypt since late January in unrest touched off by the anniversary of the uprising against Mubarak and exacerbated by a court ruling that sentenced 21 people to death over a soccer stadium disaster a year ago.

It has been the worst bloodshed since Mursi assumed office, underlining the instability that continues to thwart government efforts to restore a sense of normalcy and revive an economy in crisis by attracting fresh investment and tourism.

The unrest has been stirred by anger at Mursi and his Islamist backers, who the opposition says have betrayed the revolution and sought to monopolise power - accusations dismissed by the Muslim Brotherhood.

Recent protests have routinely turned violent, with government buildings, police stations and the presidential palace coming under fire from petrol bombs and rocks.

Trouble also flared outside of Cairo on Friday. In al-Mahalla al-Kubra, an industrial town, protesters set fire to a local government building, state TV reported. In Alexandria, the security forces briefly fought protesters when they tried to force their way into a police station, witnesses said.

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The pro-Mursi rally in Cairo grew to several thousand after prayer. "The person who came (to power) through ballot boxes will not leave by petrol bombs," said a cleric who led the crowd in Friday prayers at the rally. "He who came to power by direct, free choice will not leave by the rocks thrown by children."

"No to Violence. Yes to sharia (Islamic law)," declared a banner held aloft from the crowd.

"With our blood and souls we will sacrifice ourselves for Islam!" chanted the crowd that assembled outside Cairo University. "The people want an iron fist," read another banner.

Some people journeyed in from the provinces for the rally. At one point, the mother of Khaled al-Islambouli, one of the Islamists behind President Anwar Sadat's 1981 assassination, appeared on stage, drawing cheers as she waved to the throng.

"Mohamad Mursi is a legitimate leader who was elected by the will of the people," said Mohamad Omar, from Mansoura north of the capital. He came on a bus chartered by al-Gama'a al-Islamiya.

"The opposition's methods of violence, petrol bomb-throwing, igniting fires and causing destruction is not accepted by the Islamists," added Alaa Abdulwahab, a 45-year-old who had travelled from Minya, a four-hour journey south of Cairo.

Mursi's most prominent liberal and leftist opponents have distanced themselves from the violence, saying they support only peaceful agitation. Islamist leaders and their rivals all renounced violence at crisis talks on Jan. 31 chaired by al-Azhar, a Sunni Muslim seat of religious learning.

Islamist groups such as al-Gama'a al-Islamiya were repressed by Mubarak during his 30 years in power. But they have moved into the heart of public life since his fall, alarming secular-minded Egyptians who worry they aim to dominate the new Egypt. (Additional reporting by Mahmoud Ali; Writing by Tom Perry; Editing by Mark Heinrich)